User:Lériendil/Square and triangle superparticulars by prime subgroup: Difference between revisions

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== 2-prime subgroup (2.3) ==
Some shorthand notation used here:
* S''k'' stands for ''k''^2/[(''k''-1)(''k''+1)] by standard convention (the ''k''th square superparticular).
* T''k'' = S''k'' * S(''k''+1) stands for [''k''(''k''+1)/2]/[(''k''-1)(''k''+2)/2] (the ''k''th triangle superparticular).
* U''k'' = S''k''/S(''k''+1) stands for the ''k''th ultraparticular, which has the same subgroup as T''k'' except in the case where k is congruent to 4 (mod 9), in which case the subgroup of U''k'' lacks prime 3 from that of T''k''.
* L''p'' refers to the ''p''-limit, i.e. the subgroup of primes less than or equal to ''p''.
* L''p''(-''q'') refers to the ''p'' limit with the prime ''q'' omitted: e.g. L17(-11) refers to the 2.3.5.7.13.17 subgroup; these omissions can be stacked so that L23(-5.17) refers to the group 2.3.7.11.13.19.23.
 
This list eventually aims to be complete to the 29-add-one-limit, i.e. the class of subgroups with at most one prime greater than 29, which is a superset of the 31-limit.
 
== 2- and 3-prime subgroups (2.3 and 2.3.p) ==
Note that the following lists are ''complete'' and the insertion of higher primes will add no new inclusions to them.
 
=== 2-prime subgroup (2.3) ===


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
! rowspan="2" | Superparticular
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! rowspan="2" | Subgroup
! colspan="2" | Comma
! colspan="2" | Comma
Line 9: Line 21:
! Ratio
! Ratio
! Smonzo
! Smonzo
|-
|- style="background-color: #BFD7FF;"
| T2
| [[3-limit|2.3]]
| [[3/2]]
| {{monzo| -1 1 }}
|-
| S2
| S2
| [[3-limit|2.3]]
| [[3-limit|2.3]]
| [[4/3]]
| [[4/3]]
| {{monzo| 2 -1 }}
| {{monzo| 2 -1 }}
|-
|- style="background-color: #BFD7FF;"
| S3
| S3
| [[3-limit|2.3]]
| [[3-limit|2.3]]
Line 26: Line 33:
|}
|}


== 3-prime subgroups (2.3.p) ==
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Triangle-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Ultraparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|- style="background-color: #BFD7FF;"
| T2
| [[3-limit|2.3]]
| [[3/2]]
| {{monzo| -1 1 }}
| U2
| [[3-limit|2.3]]
| [[32/27]]
| {{monzo| 5 -3 }}
|}
 
=== 3-prime subgroups (2.3.p) ===


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
! rowspan="2" | Superparticular
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! rowspan="2" | Subgroup
! colspan="2" | Comma
! colspan="2" | Comma
Line 36: Line 67:
! Ratio
! Ratio
! Smonzo
! Smonzo
|-
|- style="background-color: #FFBFEF;"
| S4
| S4
| [[5-limit|2.3.5]]
| [[5-limit|L5]]
| [[16/15]]
| [[16/15]]
| {{monzo| 4 -1 -1 }}
| {{monzo| 4 -1 -1 }}
|-
|- style="background-color: #FFBFEF;"
| S5
| S5
| [[5-limit|2.3.5]]
| [[5-limit|L5]]
| [[25/24]]
| [[25/24]]
| {{monzo| -3 -1 2 }}
| {{monzo| -3 -1 2 }}
|-
|- style="background-color: #FFBFEF;"
| S9
| S9
| [[5-limit|2.3.5]]
| [[5-limit|L5]]
| [[81/80]]
| [[81/80]]
| {{monzo| -4 4 -1 }}
| {{monzo| -4 4 -1 }}
|-
|- style="background-color: #C7BFDF;"
| S7
| S7
| [[2.3.7 subgroup|2.3.7]]
| [[2.3.7 subgroup|2.3.7]]
| [[49/48]]
| [[49/48]]
| {{monzo| -4 -1 2 }}
| {{monzo| -4 -1 2 }}
|-
|- style="background-color: #C7BFDF;"
| S8
| S8
| [[2.3.7 subgroup|2.3.7]]
| [[2.3.7 subgroup|2.3.7]]
| [[64/63]]
| [[64/63]]
| {{monzo| 6 -2 -1 }}
| {{monzo| 6 -2 -1 }}
|-
|- style="background-color: #F7F7BF;"
| S17
| S17
| 2.3.17
| 2.3.17
| [[289/288]]
| [[289/288]]
| {{monzo| -5 -2 2 }}
| {{monzo| -5 -2 2 }}
|}
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Triangle-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Ultraparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|- style="background-color: #FFBFEF;"
| T3
| T3
| [[5-limit|2.3.5]]
| [[5-limit|L5]]
| [[6/5]]
| [[6/5]]
| {{monzo| 1 1 -1 }}
| {{monzo| 1 1 -1 }}
|-
| U3
| [[5-limit|L5]]
| [[135/128]]
| {{monzo| -7 3 1 }}
|- style="background-color: #FFBFEF;"
| T4
| T4
| [[5-limit|2.3.5]]
| [[5-limit|L5]]
| [[10/9]]
| [[10/9]]
| {{monzo| 1 -2 1 }}
| {{monzo| 1 -2 1 }}
|-
| U4
| '''2.5'''
| [[128/125]]
| {{monzo| 7 -3 }}
|- style="background-color: #C7BFDF;"
| T7
| T7
| [[2.3.7 subgroup|2.3.7]]
| [[2.3.7 subgroup|2.3.7]]
| [[28/27]]
| [[28/27]]
| {{monzo| 2 -3 1 }}
| {{monzo| 2 -3 1 }}
| U7
| [[2.3.7 subgroup|2.3.7]]
| [[1029/1024]]
| {{monzo| -10 1 3 }}
|}
== 4-prime subgroups ==
Note that the lists of triangle-particulars are ''complete'' and the insertion of higher primes will add no new inclusions to them. The lists of square particulars other than the "Higher primes" table are likewise complete.
=== 5-add-one-limit (L5.p) ===
{| class="wikitable center-1 center-2"
|-
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
|- style="background-color: #C7BFDF;"
| S6 = T8
| [[7-limit|L7]]
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
|- style="background-color: #C7BFDF;"
| S15
| [[7-limit|L7]]
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
|- style="background-color: #C7BFDF;"
| S49
| [[7-limit|L7]]
| [[2401/2400]]
| {{monzo| -5 -1 -2 4 }}
|- style="background-color: #F7C7BF;"
| S10
| L5.11
| [[100/99]]
| {{monzo| 2 -2 2 -1 }}
|- style="background-color: #F7C7BF;"
| S11
| L5.11
| [[121/120]]
| {{monzo| -3 -1 -1 2 }}
|- style="background-color: #D7BFEF;"
| S25
| L5.13
| [[625/624]]
| {{monzo| -4 -1 4 -1 }}
|- style="background-color: #D7BFEF;"
| S26
| L5.13
| [[676/675]]
| {{monzo| 2 -3 -2 2 }}
|- style="background-color: #F7F7BF;"
| S16
| L5.17
| [[256/255]]
| {{monzo| 8 -1 -1 -1 }}
|- style="background-color: #DFD7BF;"
| S19
| L5.19
| [[361/360]]
| {{monzo| -3 -2 -1 2 }}
|- style="background-color: #FFDFBF;"
| S24
| L5.23
| [[576/575]]
| {{monzo| 6 2 -2 -1 }}
|- style="background-color: #CFC7F7;"
| S31
| L5.31
| [[961/960]]
| {{monzo| -6 -1 -1 2 }}
|- style="background-color: #FFBFEF;"
| S81
| L5.41
| [[6561/6560]]
| {{monzo| -5 8 -1 -1 }}
|- style="background-color: #FFBFEF;"
| S80
| L5.79
| [[6400/6399]]
| {{monzo| 8 -4 2 -1 }}
|}
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Triangle-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Ultraparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|- style="background-color: #C7BFDF;"
| T5
| [[7-limit|L7]]
| [[15/14]]
| {{monzo| -1 1 1 -1 }}
| U5
| [[7-limit|L7]]
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
|- style="background-color: #C7BFDF;"
| T6
| [[7-limit|L7]]
| [[21/20]]
| {{monzo| -2 1 -1 1 }}
| U6
| [[7-limit|L7]]
| [[1728/1715]]
| {{monzo| 6 3 -1 -3 }}
|- style="background-color: #C7BFDF;"
| T8 = S6
| [[7-limit|L7]]
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| U8
| [[7-limit|L7]]
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
|- style="background-color: #F7C7BF;"
| T9
| L5.11
| [[45/44]]
| {{monzo| -2 2 1 -1 }}
| U9
| L5.11
| [[8019/8000]]
| {{monzo| -6 6 -3 1 }}
|- style="background-color: #F7C7BF;"
| T10
| L5.11
| [[55/54]]
| {{monzo| -1 -3 1 1 }}
| U10
| L5.11
| [[4000/3993]]
| {{monzo| 5 -1 3 -3 }}
|- style="background-color: #D7BFEF;"
| T25
| L5.13
| [[325/324]]
| {{monzo| -2 -4 2 1 }}
| U25
| L5.13
| [[140625/140608]]
| {{monzo| -6 2 6 -3 }}
|- style="background-color: #F7F7BF;"
| T16
| L5.17
| [[136/135]]
| {{monzo| 3 -3 -1 1 }}
| U16
| L5.17
| [[24576/24565]]
| {{monzo| 13 1 -1 -3 }}
|}
=== 2.3.7.p subgroups ===
{| class="wikitable center-1 center-2"
|-
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
|- style="background-color: #D7BFEF;"
| S13
| 2.3.7.13
| [[169/168]]
| {{monzo| -3 -1 -1 2 }}
|- style="background-color: #D7BFEF;"
| S27
| 2.3.7.13
| [[729/728]]
| {{monzo| -3 6 -1 -1 }}
|- style="background-color: #D7DFCF;"
| S28
| 2.3.7.29
| [[784/783]]
| {{monzo| 4 -3 2 -1 }}
|- style="background-color: #CFC7F7;"
| S63
| 2.3.7.31
| [[3969/3968]]
| {{monzo| -7 4 2 -1 }}
|- style="background-color: #C7BFDF;"
| S48
| 2.3.7.47
| [[2304/2303]]
| {{monzo| 8 2 -2 -1 }}
|- style="background-color: #C7BFDF;"
| S97
| 2.3.7.97
| [[9409/9408]]
| {{monzo| -6 -1 -2 2 }}
|- style="background-color: #C7BFDF;"
| S127
| 2.3.7.127
| [[16129/16128]]
| {{monzo| -8 -2 -1 2 }}
|}
=== 2.3.11.p subgroups ===
{| class="wikitable center-1 center-2"
|-
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
|- style="background-color: #D7BFEF;"
| S12
| 2.3.11.13
| [[144/143]]
| {{monzo| 4 2 -1 -1 }}
|- style="background-color: #F7F7BF;"
| S33
| 2.3.11.17
| [[1089/1088]]
| {{monzo| -6 2 -1 2 }}
|- style="background-color: #FFDFBF;"
| S23
| 2.3.11.23
| [[529/528]]
| {{monzo| -4 -1 -1 2 }}
|- style="background-color: #CFC7F7;"
| S32
| 2.3.11.31
| [[1024/1023]]
| {{monzo| 10 -1 -1 -1 }}
|- style="background-color: #F7C7BF;"
| S243
| 2.3.11.61
| [[59049/59048]]
| {{monzo| -3 10 -2 -1 }}
|- style="background-color: #F7C7BF;"
| S242
| 2.3.11.241
| [[58564/58563]]
| {{monzo| 2 -5 4 -1 }}
|}
=== Higher primes ===
{| class="wikitable center-1 center-2"
|-
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
|- style="background-color: #D7BFEF;"
| S53
| 2.3.13.53
| [[2809/2808]]
| {{monzo| -3 -3 -1 2 }}
|- style="background-color: #DFD7BF;"
| S18
| 2.3.17.19
| [[324/323]]
| {{monzo| 2 4 -1 -1 }}
|- style="background-color: #F7F7BF;"
| S577
| 2.3.17.577
| [[332929/332928]]
| {{monzo| -7 -2 -2 2 }}
|- style="background-color: #DFD7BF;"
| S37
| 2.3.19.37
| [[1369/1368]]
| {{monzo| -3 -2 -1 2 }}
|- style="background-color: #FFDFBF;"
| S47
| 2.3.23.47
| [[2209/2208]]
| {{monzo| -5 -1 -1 2 }}
|}
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Triangle-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Ultraparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|- style="background-color: #DFD7BF;"
| T17
| 2.3.17.19
| [[153/152]]
| {{monzo| -3 2 1 -1 }}
| U17
| 2.3.17.19
| [[93347/93312]]
| {{monzo| -7 -6 3 1 }}
|}
== 5-prime subgroups ==
=== 7-add-one-limit (L7.p) ===
{| class="wikitable center-1 center-2"
|-
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
|- style="background-color: #F7C7BF;"
| S21
| [[11-limit|L11]]
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
|- style="background-color: #F7C7BF;"
| S55
| [[11-limit|L11]]
| [[3025/3024]]
| {{monzo| -4 -3 2 -1 2 }}
|- style="background-color: #F7C7BF;"
| S99
| [[11-limit|L11]]
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
|- style="background-color: #D7BFEF;"
| S14
| L7.13
| [[196/195]]
| {{monzo| 2 -1 -1 2 -1 }}
|- style="background-color: #D7BFEF;"
| S64
| L7.13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 -1 }}
|- style="background-color: #F7F7BF;"
| S35 = T49
| L7.17
| [[1225/1224]]
| {{monzo| -3 -2 2 2 -1 }}
|- style="background-color: #F7F7BF;"
| S50
| L7.17
| [[2500/2499]]
| {{monzo| 2 -1 4 -2 -1 }}
|- style="background-color: #DFD7BF;"
| S20
| L7.19
| [[400/399]]
| {{monzo| 4 -1 2 -1 -1 }}
|- style="background-color: #FFDFBF;"
| S161
| L7.23
| [[25921/25920]]
| {{monzo| -6 -4 -1 2 2 }}
|- style="background-color: #D7DFCF;"
| S29
| L7.29
| [[841/840]]
| {{monzo| -3 -1 -1 -1 2 }}
|- style="background-color: #CFC7F7;"
| S125
| L7.31
| [[15625/15624]]
| {{monzo| -3 -2 6 -1 -1 }}
|- style="background-color: #C7BFDF;"
| S36
| L7.37
| [[1296/1295]]
| {{monzo| 4 4 -1 -1 -1 }}
|- style="background-color: #C7BFDF;"
| S41
| L7.41
| [[1681/1680]]
| {{monzo| -4 -1 -1 -1 2 }}
|- style="background-color: #C7BFDF;"
| S244
| L7.61
| [[59536/59535]]
| {{monzo| 4 -5 -1 -2 2 }}
|- style="background-color: #C7BFDF;"
| S71
| L7.71
| [[5041/5040]]
| {{monzo| -4 -2 -1 -1 2 }}
|- style="background-color: #C7BFDF;"
| S225
| L7.113
| [[50625/50624]]
| {{monzo| -6 4 -1 4 -1 }}
|- style="background-color: #C7BFDF;"
| S126
| L7.127
| [[15876/15875]]
| {{monzo| 2 4 -3 2 -1 }}
|- style="background-color: #C7BFDF;"
| S224
| L7.223
| [[50176/50175]]
| {{monzo| 10 -2 -2 2 -1 }}
|- style="background-color: #C7BFDF;"
| S251
| L7.251
| [[59536/59535]]
| {{monzo| -3 -2 -3 -1 2 }}
|- style="background-color: #C7BFDF;"
| S449
| L7.449
| [[201601/201600]]
| {{monzo| -7 -2 -2 -1 2 }}
|- style="background-color: #C7BFDF;"
| S4375
| L7.547
| [[19140625/19140624]]
| {{monzo| -4 -7 8 2 -1 }}
|- style="background-color: #C7BFDF;"
| S2401
| L7.1201
| [[5764801/5764800]]
| {{monzo| -5 -1 -2 8 -1 }}
|- style="background-color: #C7BFDF;"
| S2400
| L7.2399
| [[5760000/5759999]]
| {{monzo| 10 2 4 -4 -1 }}
|- style="background-color: #C7BFDF;"
| S4801
| L7.4801
| [[23049601/23049600]]
| {{monzo| -7 -1 -2 -4 2 }}
|}
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Triangle-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Ultraparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|- style="background-color: #D7BFEF;"
| T13
| L7.13
| [[91/90]]
| {{monzo| -1 -2 -1 1 1 }}
| U13
| '''2.5.7.13'''
| [[10985/10976]]
| {{monzo| -5 1 -3 3 }}
|- style="background-color: #D7BFEF;"
| T14
| L7.13
| [[105/104]]
| {{monzo| -3 1 1 1 -1 }}
| U14
| L7.13
| [[43904/43875]]
| {{monzo| 7 -3 -3 3 -1 }}
|- style="background-color: #D7BFEF;"
| T26
| L7.13
| [[351/350]]
| {{monzo| -1 3 -2 -1 1 }}
| U26
| L7.13
| [[492128/492075]]
| {{monzo| 5 -9 -2 1 3 }}
|- style="background-color: #F7F7BF;"
| T15
| L7.17
| [[120/119]]
| {{monzo| 3 1 1 -1 -1 }}
| U15
| L7.17
| [[57375/57344]]
| {{monzo| -13 3 3 -1 1 }}
|- style="background-color: #F7F7BF;"
| T49 = S35
| L7.17
| [[1225/1224]]
| {{monzo| -3 -2 2 2 -1 }}
| U49
| '''2.5.7.17'''
| [[2000033/2000000]]
| {{monzo| -7 -6 6 1 }}
|- style="background-color: #DFD7BF;"
| T19
| L7.19
| [[190/189]]
| {{monzo| 1 -3 1 -1 1 }}
| U19
| L7.19
| [[48013/48000]]
| {{monzo| -7 -1 -3 1 3 }}
|- style="background-color: #D7DFCF;"
| T28
| L7.29
| [[406/405]]
| {{monzo| 1 -4 -1 1 1 }}
| U28
| L7.29
| [[219520/219501]]
| {{monzo| 7 -2 1 3 -3 }}
|- style="background-color: #C7BFDF;"
| T126
| L7.127
| [[8001/8000]]
| {{monzo| -6 2 -3 1 1 }}
| U126
| L7.127
| [[256048128/256047875]]
| {{monzo| 10 6 -3 3 -3 }}
|}
=== 11-add-one-limit ===
==== L5.11.p subgroups ====
{| class="wikitable center-1 center-2"
|-
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
|- style="background-color: #D7BFEF;"
| S65
| L13(-7)
| [[4225/4224]]
| {{monzo| -7 -1 2 -1 2 }}
|- style="background-color: #FFDFBF;"
| S45
| L5.11.23
| [[2025/2024]]
| {{monzo| -3 4 2 -1 -1 }}
|- style="background-color: #F7C7BF;"
| S44
| L5.11.43
| [[1936/1935]]
| {{monzo| 4 -2 -1 2 -1 }}
|- style="background-color: #F7C7BF;"
| S54
| L5.11.53
| [[2916/2915]]
| {{monzo| 2 6 -1 -1 -1 }}
|- style="background-color: #F7C7BF;"
| S121
| L5.11.61
| [[14641/14640]]
| {{monzo| -4 -1 -1 4 -1 }}
|- style="background-color: #F7C7BF;"
| S89
| L5.11.89
| [[7921/7920]]
| {{monzo| -4 -2 -1 -1 2 }}
|- style="background-color: #F7C7BF;"
| S485
| L5.11.97
| [[235225/235224]]
| {{monzo| -3 -5 2 -2 2 }}
|- style="background-color: #F7C7BF;"
| S100
| L5.11.101
| [[10000/9999]]
| {{monzo| 4 -2 4 -1 -1 }}
|- style="background-color: #F7C7BF;"
| S109
| L5.11.109
| [[11881/11880]]
| {{monzo| -3 -3 -1 -1 2 }}
|- style="background-color: #F7C7BF;"
| S199
| L5.11.199
| [[39601/39600]]
| {{monzo| -4 -2 -2 -1 2 }}
|- style="background-color: #F7C7BF;"
| S241
| L5.11.241
| [[58081/58080]]
| {{monzo| -5 -1 -1 -2 2 }}
|}
|}
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Triangle-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Ultraparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|- style="background-color: #D7BFEF;"
| T11
| L13(-7)
| [[66/65]]
| {{monzo| 1 1 -1 1 -1 }}
| U11
| L13(-7)
| [[17303/17280]]
| {{monzo| 7 3 1 -3 -1 }}
|- style="background-color: #FFDFBF;"
| T23
| L5.11.23
| [[276/275]]
| {{monzo| 2 1 -2 -1 1 }}
| U23
| L5.11.23
| [[304175/304128]]
| {{monzo| -10 -3 2 -1 3 }}
|- style="background-color: #CFC7F7;"
| T31
| L5.11.31
| [[496/495]]
| {{monzo| 4 -2 -1 -1 1 }}
| U31
| '''2.5.11.31'''
| [[327701/327680]]
| {{monzo| -16 -1 1 3 }}
|- style="background-color: #F7C7BF;"
| T241
| L5.11.241
| [[29161/29160]]
| {{monzo| -4 -6 -1 2 1 }}
| U241
| L5.11.241
| [[1133799201/1133799040]]
| {{monzo| -7 4 -1 -6 3 }}
|}
==== No-fives (L11(-5).p) subgroups ====
{| class="wikitable center-1 center-2"
|-
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
|- style="background-color: #FFDFBF;"
| S22
| L11(-5).23
| [[484/483]]
| {{monzo| 2 -1 -1 2 -1 }}
|- style="background-color: #F7C7BF;"
| S43
| L11(-5).43
| [[1849/1848]]
| {{monzo| -3 -1 -1 -1 2 }}
|- style="background-color: #F7C7BF;"
| S98
| L11(-5).97
| [[9604/9603]]
| {{monzo| 2 -2 4 -1 -1 }}
|- style="background-color: #F7C7BF;"
| S197
| L11(-5).197
| [[38809/38808]]
| {{monzo| -3 -2 -2 -1 2 }}
|}
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Triangle-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Ultraparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|- style="background-color: #D7BFEF;"
| T12
| L13(-5)
| [[78/77]]
| {{monzo| 1 1 -1 -1 1 }}
| U12
| L13(-5)
| [[24192/24167]]
| {{monzo| 7 3 1 -1 -3 }}
|- style="background-color: #FFDFBF;"
| T22
| L11(-5).23
| [[253/252]]
| {{monzo| -2 -2 -1 1 1 }}
| U22
| '''2.7.11.23'''
| [[85184/85169]]
| {{monzo| 6 -1 3 -3 }}
|- style="background-color: #F7C7BF;"
| T97
| L11(-5).97
| [[4753/4752]]
| {{monzo| -4 -3 2 -1 1 }}
| U97
| L11(-5).97
| [[30118209/30118144]]
| {{monzo| -8 1 -6 1 3 }}
|}
== See also ==
* [[User:Lériendil/Third-superparticulars_and_semiparticulars_by_prime_subgroup|Third-superparticulars and semiparticulars by prime subgroup]]

Latest revision as of 18:12, 28 July 2024

Some shorthand notation used here:

  • Sk stands for k^2/[(k-1)(k+1)] by standard convention (the kth square superparticular).
  • Tk = Sk * S(k+1) stands for [k(k+1)/2]/[(k-1)(k+2)/2] (the kth triangle superparticular).
  • Uk = Sk/S(k+1) stands for the kth ultraparticular, which has the same subgroup as Tk except in the case where k is congruent to 4 (mod 9), in which case the subgroup of Uk lacks prime 3 from that of Tk.
  • Lp refers to the p-limit, i.e. the subgroup of primes less than or equal to p.
  • Lp(-q) refers to the p limit with the prime q omitted: e.g. L17(-11) refers to the 2.3.5.7.13.17 subgroup; these omissions can be stacked so that L23(-5.17) refers to the group 2.3.7.11.13.19.23.

This list eventually aims to be complete to the 29-add-one-limit, i.e. the class of subgroups with at most one prime greater than 29, which is a superset of the 31-limit.

2- and 3-prime subgroups (2.3 and 2.3.p)

Note that the following lists are complete and the insertion of higher primes will add no new inclusions to them.

2-prime subgroup (2.3)

Square-particular Subgroup Comma
Ratio Smonzo
S2 2.3 4/3 [2 -1⟩
S3 2.3 9/8 [-3 2⟩
Triangle-particular Subgroup Comma Ultraparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
T2 2.3 3/2 [-1 1⟩ U2 2.3 32/27 [5 -3⟩

3-prime subgroups (2.3.p)

Square-particular Subgroup Comma
Ratio Smonzo
S4 L5 16/15 [4 -1 -1⟩
S5 L5 25/24 [-3 -1 2⟩
S9 L5 81/80 [-4 4 -1⟩
S7 2.3.7 49/48 [-4 -1 2⟩
S8 2.3.7 64/63 [6 -2 -1⟩
S17 2.3.17 289/288 [-5 -2 2⟩
Triangle-particular Subgroup Comma Ultraparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
T3 L5 6/5 [1 1 -1⟩ U3 L5 135/128 [-7 3 1⟩
T4 L5 10/9 [1 -2 1⟩ U4 2.5 128/125 [7 -3⟩
T7 2.3.7 28/27 [2 -3 1⟩ U7 2.3.7 1029/1024 [-10 1 3⟩

4-prime subgroups

Note that the lists of triangle-particulars are complete and the insertion of higher primes will add no new inclusions to them. The lists of square particulars other than the "Higher primes" table are likewise complete.

5-add-one-limit (L5.p)

Square-particular Subgroup Comma
Ratio Smonzo
S6 = T8 L7 36/35 [2 2 -1 -1⟩
S15 L7 225/224 [-5 2 2 -1⟩
S49 L7 2401/2400 [-5 -1 -2 4⟩
S10 L5.11 100/99 [2 -2 2 -1⟩
S11 L5.11 121/120 [-3 -1 -1 2⟩
S25 L5.13 625/624 [-4 -1 4 -1⟩
S26 L5.13 676/675 [2 -3 -2 2⟩
S16 L5.17 256/255 [8 -1 -1 -1⟩
S19 L5.19 361/360 [-3 -2 -1 2⟩
S24 L5.23 576/575 [6 2 -2 -1⟩
S31 L5.31 961/960 [-6 -1 -1 2⟩
S81 L5.41 6561/6560 [-5 8 -1 -1⟩
S80 L5.79 6400/6399 [8 -4 2 -1⟩
Triangle-particular Subgroup Comma Ultraparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
T5 L7 15/14 [-1 1 1 -1⟩ U5 L7 875/864 [-5 -3 3 1⟩
T6 L7 21/20 [-2 1 -1 1⟩ U6 L7 1728/1715 [6 3 -1 -3⟩
T8 = S6 L7 36/35 [2 2 -1 -1⟩ U8 L7 5120/5103 [10 -6 1 -1⟩
T9 L5.11 45/44 [-2 2 1 -1⟩ U9 L5.11 8019/8000 [-6 6 -3 1⟩
T10 L5.11 55/54 [-1 -3 1 1⟩ U10 L5.11 4000/3993 [5 -1 3 -3⟩
T25 L5.13 325/324 [-2 -4 2 1⟩ U25 L5.13 140625/140608 [-6 2 6 -3⟩
T16 L5.17 136/135 [3 -3 -1 1⟩ U16 L5.17 24576/24565 [13 1 -1 -3⟩

2.3.7.p subgroups

Square-particular Subgroup Comma
Ratio Smonzo
S13 2.3.7.13 169/168 [-3 -1 -1 2⟩
S27 2.3.7.13 729/728 [-3 6 -1 -1⟩
S28 2.3.7.29 784/783 [4 -3 2 -1⟩
S63 2.3.7.31 3969/3968 [-7 4 2 -1⟩
S48 2.3.7.47 2304/2303 [8 2 -2 -1⟩
S97 2.3.7.97 9409/9408 [-6 -1 -2 2⟩
S127 2.3.7.127 16129/16128 [-8 -2 -1 2⟩

2.3.11.p subgroups

Square-particular Subgroup Comma
Ratio Smonzo
S12 2.3.11.13 144/143 [4 2 -1 -1⟩
S33 2.3.11.17 1089/1088 [-6 2 -1 2⟩
S23 2.3.11.23 529/528 [-4 -1 -1 2⟩
S32 2.3.11.31 1024/1023 [10 -1 -1 -1⟩
S243 2.3.11.61 59049/59048 [-3 10 -2 -1⟩
S242 2.3.11.241 58564/58563 [2 -5 4 -1⟩

Higher primes

Square-particular Subgroup Comma
Ratio Smonzo
S53 2.3.13.53 2809/2808 [-3 -3 -1 2⟩
S18 2.3.17.19 324/323 [2 4 -1 -1⟩
S577 2.3.17.577 332929/332928 [-7 -2 -2 2⟩
S37 2.3.19.37 1369/1368 [-3 -2 -1 2⟩
S47 2.3.23.47 2209/2208 [-5 -1 -1 2⟩
Triangle-particular Subgroup Comma Ultraparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
T17 2.3.17.19 153/152 [-3 2 1 -1⟩ U17 2.3.17.19 93347/93312 [-7 -6 3 1⟩

5-prime subgroups

7-add-one-limit (L7.p)

Square-particular Subgroup Comma
Ratio Smonzo
S21 L11 441/440 [-3 2 -1 2 -1⟩
S55 L11 3025/3024 [-4 -3 2 -1 2⟩
S99 L11 9801/9800 [-3 4 -2 -2 2⟩
S14 L7.13 196/195 [2 -1 -1 2 -1⟩
S64 L7.13 4096/4095 [12 -2 -1 -1 -1⟩
S35 = T49 L7.17 1225/1224 [-3 -2 2 2 -1⟩
S50 L7.17 2500/2499 [2 -1 4 -2 -1⟩
S20 L7.19 400/399 [4 -1 2 -1 -1⟩
S161 L7.23 25921/25920 [-6 -4 -1 2 2⟩
S29 L7.29 841/840 [-3 -1 -1 -1 2⟩
S125 L7.31 15625/15624 [-3 -2 6 -1 -1⟩
S36 L7.37 1296/1295 [4 4 -1 -1 -1⟩
S41 L7.41 1681/1680 [-4 -1 -1 -1 2⟩
S244 L7.61 59536/59535 [4 -5 -1 -2 2⟩
S71 L7.71 5041/5040 [-4 -2 -1 -1 2⟩
S225 L7.113 50625/50624 [-6 4 -1 4 -1⟩
S126 L7.127 15876/15875 [2 4 -3 2 -1⟩
S224 L7.223 50176/50175 [10 -2 -2 2 -1⟩
S251 L7.251 59536/59535 [-3 -2 -3 -1 2⟩
S449 L7.449 201601/201600 [-7 -2 -2 -1 2⟩
S4375 L7.547 19140625/19140624 [-4 -7 8 2 -1⟩
S2401 L7.1201 5764801/5764800 [-5 -1 -2 8 -1⟩
S2400 L7.2399 5760000/5759999 [10 2 4 -4 -1⟩
S4801 L7.4801 23049601/23049600 [-7 -1 -2 -4 2⟩
Triangle-particular Subgroup Comma Ultraparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
T13 L7.13 91/90 [-1 -2 -1 1 1⟩ U13 2.5.7.13 10985/10976 [-5 1 -3 3⟩
T14 L7.13 105/104 [-3 1 1 1 -1⟩ U14 L7.13 43904/43875 [7 -3 -3 3 -1⟩
T26 L7.13 351/350 [-1 3 -2 -1 1⟩ U26 L7.13 492128/492075 [5 -9 -2 1 3⟩
T15 L7.17 120/119 [3 1 1 -1 -1⟩ U15 L7.17 57375/57344 [-13 3 3 -1 1⟩
T49 = S35 L7.17 1225/1224 [-3 -2 2 2 -1⟩ U49 2.5.7.17 2000033/2000000 [-7 -6 6 1⟩
T19 L7.19 190/189 [1 -3 1 -1 1⟩ U19 L7.19 48013/48000 [-7 -1 -3 1 3⟩
T28 L7.29 406/405 [1 -4 -1 1 1⟩ U28 L7.29 219520/219501 [7 -2 1 3 -3⟩
T126 L7.127 8001/8000 [-6 2 -3 1 1⟩ U126 L7.127 256048128/256047875 [10 6 -3 3 -3⟩

11-add-one-limit

L5.11.p subgroups

Square-particular Subgroup Comma
Ratio Smonzo
S65 L13(-7) 4225/4224 [-7 -1 2 -1 2⟩
S45 L5.11.23 2025/2024 [-3 4 2 -1 -1⟩
S44 L5.11.43 1936/1935 [4 -2 -1 2 -1⟩
S54 L5.11.53 2916/2915 [2 6 -1 -1 -1⟩
S121 L5.11.61 14641/14640 [-4 -1 -1 4 -1⟩
S89 L5.11.89 7921/7920 [-4 -2 -1 -1 2⟩
S485 L5.11.97 235225/235224 [-3 -5 2 -2 2⟩
S100 L5.11.101 10000/9999 [4 -2 4 -1 -1⟩
S109 L5.11.109 11881/11880 [-3 -3 -1 -1 2⟩
S199 L5.11.199 39601/39600 [-4 -2 -2 -1 2⟩
S241 L5.11.241 58081/58080 [-5 -1 -1 -2 2⟩
Triangle-particular Subgroup Comma Ultraparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
T11 L13(-7) 66/65 [1 1 -1 1 -1⟩ U11 L13(-7) 17303/17280 [7 3 1 -3 -1⟩
T23 L5.11.23 276/275 [2 1 -2 -1 1⟩ U23 L5.11.23 304175/304128 [-10 -3 2 -1 3⟩
T31 L5.11.31 496/495 [4 -2 -1 -1 1⟩ U31 2.5.11.31 327701/327680 [-16 -1 1 3⟩
T241 L5.11.241 29161/29160 [-4 -6 -1 2 1⟩ U241 L5.11.241 1133799201/1133799040 [-7 4 -1 -6 3⟩

No-fives (L11(-5).p) subgroups

Square-particular Subgroup Comma
Ratio Smonzo
S22 L11(-5).23 484/483 [2 -1 -1 2 -1⟩
S43 L11(-5).43 1849/1848 [-3 -1 -1 -1 2⟩
S98 L11(-5).97 9604/9603 [2 -2 4 -1 -1⟩
S197 L11(-5).197 38809/38808 [-3 -2 -2 -1 2⟩
Triangle-particular Subgroup Comma Ultraparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
T12 L13(-5) 78/77 [1 1 -1 -1 1⟩ U12 L13(-5) 24192/24167 [7 3 1 -1 -3⟩
T22 L11(-5).23 253/252 [-2 -2 -1 1 1⟩ U22 2.7.11.23 85184/85169 [6 -1 3 -3⟩
T97 L11(-5).97 4753/4752 [-4 -3 2 -1 1⟩ U97 L11(-5).97 30118209/30118144 [-8 1 -6 1 3⟩

See also

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